The problem is asking to demonstrate two properties of definite integrals related to even and odd functions.

Solution

Part (a)

Statement: Show that if $f$ is an even function, then $\int_{-a}^{a} f(x) \, dx = 2 \int_0^{a} f(x) \, dx.$

1. Definition of an even function: A function $f(x)$ is even if $f(-x) = f(x)$ for all $x$ in its domain.

2. Set up the integral: $\int_{-a}^{a} f(x) \, dx = \int_{-a}^{0} f(x) \, dx + \int_{0}^{a} f(x) \, dx.$

3. Use the property of even functions: Since $f(x)$ is even, $f(x) = f(-x)$. In the interval $[-a, 0]$, let $u = -x$ (then $du = -dx$). When $x = -a$, $u = a$, and when $x = 0$, $u = 0$. Thus, $\int_{-a}^{0} f(x) \, dx = \int_{a}^{0} f(-u) \, (-du) = \int_{0}^{a} f(u) \, du = \int_{0}^{a} f(x) \, dx.$

4. Combine integrals: $\int_{-a}^{a} f(x) \, dx = \int_{0}^{a} f(x) \, dx + \int_{0}^{a} f(x) \, dx = 2 \int_{0}^{a} f(x) \, dx.$ Thus, the result is proven.

Part (b)

Statement: Show that if $f$ is an odd function, then $\int_{-a}^{a} f(x) \, dx = 0.$

1. Definition of an odd function: A function $f(x)$ is odd if $f(-x) = -f(x)$ for all $x$ in its domain.

2. Set up the integral: $\int_{-a}^{a} f(x) \, dx = \int_{-a}^{0} f(x) \, dx + \int_{0}^{a} f(x) \, dx.$

3. Use the property of odd functions: Since $f(x)$ is odd, $f(-x) = -f(x)$. In the interval $[-a, 0]$, let $u = -x$ (then $du = -dx$). When $x = -a$, $u = a$, and when $x = 0$, $u = 0$. Thus, $\int_{-a}^{0} f(x) \, dx = \int_{a}^{0} f(-u) \, (-du) = -\int_{0}^{a} f(u) \, du = -\int_{0}^{a} f(x) \, dx.$

4. Combine integrals: $\int_{-a}^{a} f(x) \, dx = \int_{0}^{a} f(x) \, dx - \int_{0}^{a} f(x) \, dx = 0.$ Thus, the result is proven.

Would you like more details on any of these steps, or do you have further questions?

Here are 5 related questions to expand your understanding:

1. What are some examples of even functions and odd functions?
2. How would these properties change if $f(x)$ were neither even nor odd?
3. Why is the integral of an odd function over a symmetric interval always zero?
4. Can you apply these properties to solve definite integrals more efficiently?
5. How do these properties apply in Fourier analysis for decomposing functions?

Tip: When calculating definite integrals over symmetric intervals, check if the function is even or odd. This can simplify the calculation significantly.